Digital Communication System

ABSTRACT

Methods and apparatus for use in communication systems using recursive modulation schemes with a Low Density Generator Matrix code (including an irregular repeat accumulate (IRA) code) are described that have reduced complexity and thus reduced cost compared to prior art systems. A communication system is described in which the transmitter concatenates a low density generator matrix code with an accumulator followed by a recursive modulator in order to eliminate the use of an interleaver, and in which the receiver combines the decoder for the accumulator and the soft demodulator into a single joint decoder in order reduce the number of components and complexity. Another variation is also described in which the transmitter is further simplified by eliminated the accumulator altogether, and in which the receiver is further simplified by replacing the joint decoder with a soft demodulator prior to the LDGM soft decoder.

INCORPORATION BY REFERENCE

Co-pending PCT Patent Application (number yet to be assigned) titled“Carrier phase and amplitude estimation for phase shift keying usingpilots and data” filed on 13 Dec. 2013 is referred to in the followingdescription, the content of which is hereby incorporated by reference inits entirety.

The following co-pending PCT applications are referred to in thefollowing description:

PCT/AU2013/000888 titled “System and Method for Analog to DigitalConversion” filed on 13 Aug. 2013;

PCT/AU2013/000895 titled “Channel Allocation in a Communication System”filed on 14 Aug. 2013;

PCT/AU2013/001078 titled “Communication system and method” filed on 20Sep. 2013; and

PCT/AU2013/001079 titled “Multi-Access Communication System” filed on 20Sep. 2013.

The content of each of these applications is hereby incorporated byreference in their entirety.

PRIORITY DOCUMENTS

The present application claims priority from Australian ProvisionalPatent Application No. 2012905588 titled “DIGITAL COMMUNICATION SYSTEM”and filed on 20 Dec. 2012, the content of which is hereby incorporatedby reference in its entirety.

TECHNICAL FIELD

The present invention relates to digital communication systems. In aparticular form the present invention relates to digital communicationsystems using recursive modulation schemes.

BACKGROUND

Many digital modulation schemes in current use are memoryless, meaningthat the modulated signal within a particular time interval (called thesymbol duration), corresponding to a particular data bit or group ofdata bits, is independent of the modulated signal for all other symboldurations. Some well-known examples of memoryless modulation schemesinclude binary phase shift keying (BPSK), quaternary phase shift keying(QPSK), and quadrature amplitude modulation (QAM). In this context amodulation scheme refers to the modulation and correspondingdemodulation protocols or methods which are implemented in modulatorsand demodulators.

In contrast, a modulation scheme with memory produces a modulated signalthat depends not only on the current bit (or group of bits), but also onprevious bits or groups of bits. In this case the modulators anddemodulators include a memory or register for storing previouslytransmitted or received bits or symbols. One particular class ofmodulation schemes with memory (ie non-memoryless) are recursivemodulation schemes. These are modulation schemes with memory, definedrecursively. Recursive modulators are characterised by their use offeedback within the modulator structure, in which the modulator outputdepends not only on current and previous data bits, but also on previousmodulator outputs (or more generally on previous values of internalstate variables). Notable examples of recursive modulation schemes aredifferential modulation in which the data is modulated onto differencesbetween successive symbols, and continuous phase modulation where arecursive modulator is used to ensure constant amplitude and continuousphase properties. For example in differential modulation a transmittedbit is obtained by binary (ie mod 2) addition of the previouslytransmitted bit and the bit to be transmitted (ie y_(i)=y_(i−1)⊕x_(i)).Similarly on the receive (or decoding) side, the bit to be decoded isthe binary addition of the received bit and the previously received bit(ie x_(i)=y_(i)⊕y_(i−1)).

Differential modulation schemes, such as differential phase shift keying(DPSK), are an attractive choice for situations where the absolute phaseof the received signal is unknown, or is difficult to recover. The datais modulated onto the phase difference between successive symbols,rather than an absolute phase. Note that all differential modulationschemes are recursive (that is their modulator includes a feedbackloop).

Advantages of differential modulation schemes include that it ispossible to implement a non-coherent demodulator that does not need toperform computationally expensive carrier phase recovery. This cangreatly decrease the implementation complexity of the receiver. Anotheradvantage is that only a modest change is required at the transmitter toachieve the differential modulation. Disadvantages of differentialmodulation include that a single symbol erasure can affect thesubsequent symbol(s), since the demodulator needs to know the phasedifference between adjacent symbols. A further disadvantage is a 3 dBloss in performance when non-coherent demodulation is used.

Many variations of differential modulation schemes are possible, byvarying the set of possible phase changes, and by varying the mappingbetween bits and phase changes. Differential modulation schemes are usedin a wide variety of wired, radio frequency wireless and opticalwireless communication systems. Common examples include differentialbinary phase shift keying (DBPSK), differential quaternary phase shiftkeying (DQPSK), or other variants such as π/4 offset differentialquaternary phase shift keying (π/4-DQPSK).

In many circumstances it may be desirable to use a continuous phasemodulation scheme (CPM). CPM provides higher spectral efficiencycompared to other modulation schemes such as Phase Shift Keying (PSK)and a constant modulus allows the use of lower cost amplifiers sinceless headroom needs to be supplied. However there are also disadvantagesassociated with CPM. These include a higher complexity demodulator andthe potential for catastrophic error propagation. For example if onesymbol is “erased” all subsequent symbols are also erased. CPM is usedin a wide range of systems including cellular communications systemssuch as GSM, automatic identification system (AIS), satellitecommunications, and various others. One widely used version of CPM isGaussian minimum shift keying (GMSK).

It is desirable to increase the reliability of digital communicationsystems by using forward error control coding. The general advantagesinclude increased spectral efficiency, increased power efficiency, andgreatly decreased bit error rate (BER) or word error rate of the decodedsignal. Modern error control codes such as turbo codes or low densityparity check codes can offer performance that approach fundamentallimits set by information theory.

One modern code which offers very good performance and has severalimplementation advantages is the irregular repeat accumulate (IRA) code.These codes have a low-complexity encoder implementation and a lowcomplexity iterative decoding algorithm. These codes can approach theShannon capacity of many channels.

FIG. 1 is a functional block diagram for implementing an IrregularRepeat Accumulate (IRA) code 10 by an encoder module in a transmitter.As shown in FIG. 1, an IRA code consists of the serial concatenation ofan irregular repeat code encoder module 1, an interleaver module 2, aparity check module 3 and an accumulator module 5. As shown in FIG. 1,the IRA encoder receives a plurality of source bits u=(u₁ . . . , u_(k))6 and generates a plurality of encoded bits x=(x₁, . . . , x_(n)) 9. Thecombination of the irregular repeat code, interleaver and irregularparity checks can be viewed without loss of generality as a low densitygenerator matrix (LDGM) code 4. By itself however, an LDGM code offerspoor performance due to the existence of very low weight code-words.Concatenation of the outer LDGM encoder with a non-recursive inner codereplacing the accumulator would also give very poor performance.

The key to the high performance of the IRA code 10 is the recursivenature of the accumulator 5 which combines 7 adjacent coded symbols (orencoded bits) d 8 to generate encoded bits x 9 for transmission. Thisensures convergence of an iterative decoder which passes softinformation between two component decoders (a) the LDGM decoder and (b)the accumulator decoder. The output of an irregular repeat accumulatecode could be used to modulate the baseband channel input, using anysuitable modulation scheme such as PSK, PAM, QAM, DPSK, DQPSK or CPM.However, if recursive (ie non-memoryless) modulation schemes are used(eg DPSK, DQPSK or CPM), it is standard practice to insert an additionalinterleaver 11 at the output of the accumulator 5 and prior to themodulator 12. This is shown in FIG. 2.

The additional interleaver 11 is required to ensure convergence of adecoder 20 that now iterates, passing soft information between threecomponent decoders: (a) the variable node decoder 18; (b) the checknode/accumulator decoder 16; and (c) a soft demodulator 14. This decoderstructure is shown in FIG. 2 which is a functional block diagram of atransmitter and receiver for encoding and decoding an IRA code withrecursive modulation. The addition of the interleaver 11 adds to thecomplexity and cost of the transmitter and receiver (which is requiredto include a corresponding de-interleaver, marked as blocks 15 and 17 inFIG. 2) in such systems. An additional consideration with this decoderstructure is the order of activation of the three different componentdecoders. This adds to the complexity of the decoder, and may requireadditional control logic in order to optimise the order of activation.Alternatively, a fixed activation schedule could be used at the expenseof decreased performance.

These requirements add additional cost and complexity and thus there isa need to develop coding methods and apparatus for systems using LDGMcodes with recursive modulation schemes that are simpler and cheaper toimplement, or to at least provide a useful alternative to existingmethods and apparatus.

SUMMARY

According to a first aspect, there is provided an apparatus forgenerating a signal for transmission in a communication system using arecursive modulation scheme and a Low Density Generator Matrix (LDGM)code, the apparatus comprising:

an encoder for receiving a plurality of source bits and generating aplurality of coded symbols, wherein the encoder includes a Low DensityGenerator Matrix (LDGM) encoder to generate a plurality of coded symbolsaccording to a LDGM code; and

a recursive modulator which receives the plurality of coded symbols fromthe encoder and modulates the received plurality of coded symbols to asignal for transmission using a recursive modulation scheme.

In a further form, the encoder is an irregular repeat accumulate codeencoder comprising the LDGM code (serially) concatenated with anaccumulator and the plurality of coded symbols are encoded according toan IRA code. The recursive modulator may use a differential encodingmodulation scheme or a continuous phase modulation scheme. The recursivemodulator may use a finite state machine implementation.

According to a second aspect, there is provided an apparatus forreceiving a transmission in a communication system using a recursivemodulation scheme and a Low Density Generator Matrix (LDGM) code, theapparatus comprising:

a soft demodulator for receiving a transmit signal and demodulating thereceived signal according to a recursive modulation scheme to generate aplurality of coded symbol estimates; and

a Low Density Generator Matrix (LDGM) decoder for receiving theplurality of coded symbol estimates from the soft demodulator anddecoding the coded symbol estimates to generate a plurality Of sourcebit estimates.

The coded symbol estimates and source bit estimates may be hard or softestimates. In a further form, iterative decoding of the received signalis implemented between the soft demodulator and the LDGM encoder until astopping criterion is reached. Each iteration comprises two steps. Inthe first step the soft demodulator receives a plurality of softa-priori estimates of the coded symbol estimates from the LDGM decoder,and generates updated coded symbol estimates which are transmitted tothe LDGM decoder. In the second step the LDGM decoder receives aplurality of soft a-priori symbol estimates from the soft demodulatorand generates updated coded symbol estimates which are transmitted tothe soft demodulator for use as soft a-priori coded symbol estimates inthe next iteration.

In a further form, the soft demodulator further comprises an accumulatordecoder (ie jointly performs, decoding of the recursive modulation andaccumulator steps on the transmit side). In one form, the communicationsystem is a multi-access system and the soft demodulator furthergenerates a soft estimate of the transmitted signal. In another form,the soft demodulator implements a differential demodulation scheme. Inanother form, the soft demodulator, implements a continuous phasedemodulation scheme. The soft demodulator may use a trellis basedalgorithm. In one form, the soft demodulator correlates the receivedsignal with all possible transmitted waveforms over a symbol period inwhich the set of all possible transmitted waveforms is determined usingthe recursive modulation scheme implemented.

A communication system comprising a transmitter of the first aspect anda receiver of the second aspect may also be provided.

According to a third aspect, there is provided a method for generating asignal for transmission in a communication system using a recursivemodulation scheme and a Low Density Generator Matrix (LDGM) code, themethod comprising:

encoding a plurality of received source bits and generating a pluralityof coded symbols, wherein the encoder includes Low Density GeneratorMatrix (LDGM) encoder to generate a plurality of coded symbols accordingto a LDGM code; and

modulating the plurality of coded symbols using a recursive modulationscheme to generate a signal for transmission.

According to a fourth aspect, there is provided a method for receiving atransmission in a communication system using a recursive modulationscheme and a Low Density Generator Matrix (LDGM) code, the methodcomprising:

soft demodulating a received transmit signal according to a recursivemodulation scheme to generate a plurality of coded symbol estimates; and

decoding the plurality of coded symbol estimates using a Low DensityGenerator Matrix (LDGM) decoding scheme and generating a plurality ofsource bit estimates.

BRIEF DESCRIPTION OF DRAWINGS

A preferred embodiment of the present invention will be discussed withreference to the accompanying drawings wherein:

FIG. 1 is functional block diagram for implementing an Irregular RepeatAccumulate (IRA) coding scheme;

FIG. 2 is a functional block diagram of a transmitter and receiver forencoding and decoding an IRA code with a recursive modulation scheme;

FIG. 3 is a functional block diagram of a transmitter and receiver forencoding and decoding an IRA code with a recursive modulation schemeaccording to an embodiment;

FIG. 4 is a functional block diagram of a transmitter and receiver forencoding and decoding a Low Density Generator Matrix (LDGM) code with arecursive modulation scheme according to one embodiment;

FIG. 5 is a functional block diagram of a soft demodulator forgeneration of soft estimate of a transmitted signal according to anembodiment;

FIG. 6 is a functional block diagram of a transmitter and iterativereceiver for encoding and decoding a Low Density Generator Matrix (LDGM)code with a recursive modulation scheme according to one embodiment;

FIG. 7 is a finite state machine representation of a GMSK modulatoraccording to an embodiment;

FIG. 8 is a finite state machine representation of a Joint Accumulatorand GMSK Modulator according to an embodiment;

FIG. 9 is a finite state machine representation of a Differential BinaryPhase Shift Keying (DBPSK) modulator according to an embodiment;

FIG. 10 is a trellis representation of DBPSK according to an embodiment;

FIG. 11 is a finite state machine representation of a DifferentialQuaternary Phase Shift Keying (DQPSK) modulator according to anembodiment;

FIG. 12 is a trellis representation of DQPSK according to an embodiment;

FIG. 13 is a finite state machine representation of a π/4 offset DBPSKmodulator according to an embodiment;

FIG. 14 is a trellis representation of π/4 offset DBPSK according to anembodiment;

FIG. 15 is a trellis representation of a joint accumulator/DBPSK;

FIG. 16 is a figure comparing bit error rates for uncoded GMSK,LDGM-coded GMSK without iterations between the GMSK demodulator and LDGMdecoder, and LDGM-coded GMSK with iterations between GMSK demodulatorand LDGM decoder;

FIG. 17 is a figure comparing mean square error (MSE) between atransmitted LDGM/GMSK signal and a remodulated signal for variousremodulation schemes including a non-iterative receiver (ie withoutiterations between the demodulator and decoder) which makes harddecisions or soft decisions on transmitted data bits, and an iterativereceiver (ie with iterations between the demodulator and decoder) whichmakes hard decisions or soft decisions on transmitted data bits.

FIG. 18 is a flowchart of a method for generating a signal fortransmission in a communication system using a recursive modulationscheme and a Low Density Generator Matrix (LDGM) code; and

FIG. 19 is a flowchart of a method for receiving a transmission in acommunication system using a recursive modulation scheme and a LowDensity Generator Matrix (LDGM) code.

In the following description, like reference characters designate likeor corresponding parts throughout the figures.

DESCRIPTION OF EMBODIMENTS

Various embodiments of methods, apparatus and systems for generating anddecoding signals using a low density generator matrix (LDGM) code(including IRA code) with a recursive modulation (ie non-memoryless)scheme will now be described. These embodiments avoid the standardrequirement for interleavers 11 and 15 between the encoders and decodersand a three component decoder, and thus have reduced complexity and costcompared to existing systems which use an irregular repeat accumulate(IRA) code with a recursive modulation scheme such as those shown in thePrior Art system illustrated in FIGS. 1 and 2.

In one embodiment, a system using a combined accumulator and recursivemodulator has been developed. A functional block diagram of atransmitter and receiver for encoding and decoding an IRA code withrecursive modulation is illustrated in FIG. 3. In this embodiment, thetransmitter comprises a LDGM encoder module 4, followed by anaccumulator module 5 to form an IRA encoder 10, and this is concatenatedor serially connected with a recursive modulator 21. The receivercomprises a joint recursive modulator/accumulator soft decoder 22 (jointaccumulator decoder and recursive demodulator, or more concisely a jointdecoder) and a LDGM soft decoder 25. In comparison with the prior artsystem illustrated in FIG. 2, no interleaver is used between theaccumulator 5 and recursive modulator 21, and decoding is accomplishedby iterating between two components: (a) the LDGM (soft) decoder 25 and(b) a joint (soft) decoder 22 for the accumulator/modulator. Byeliminating the additional interleaver 11 on the transmit side, and bycombining the decoder for the accumulator 16 and the soft demodulator 14in the receiver into a single joint decoder (ie 22 replaces 14 and 16,and there is no need for interleaver 15) the number of components andthus complexity can be reduced.

As illustrated in FIG. 3, the encoder receives a plurality of sourcebits u 6 and generates a plurality of coded symbols (or encoded bits) d8 according to an LDGM or, after an accumulator stage, an IRA code. Therecursive modulator 21 and associated transmitter hardware (not shown)is used to generate a signal for transmission x(t) 9, which istransmitted over a (noisy) channel 13 to a receiver. The signal isattenuated/distorted by the channel and so the receiver thus sees asignal y(t) 19. The received signal 19 is passed to the joint decoder 22where iterative decoding with the LDGM decoder 25 is performed. Atiteration l, the joint decoder 22 for the recursivemodulator/accumulator takes as input the received signal y(t) 19 and aplurality of initial values (or prior values) of {circumflex over(d)}_(k) ^((l)) 24 (at l=0 these are initialised to zero, ie uniformpriors on the code bits) and computes soft bits {tilde over (d)}_(k)^((l)) 23. This joint decoder 22 can be implemented using a special formof the forward-backward algorithm operating on a joint trellis for theaccumulator 5 and recursive modulator 21. In use the encoder will beprovided with a stream (ie a plurality) of source bits 6 and generate asignal for transmission x(t) which is transmitted 9 and then decoded togenerate an output stream (ie a plurality) of estimates of the sourcebits 26.

An example for use with a GMSK (ie CPM recursive modulation) scheme isdiscussed below, along with an example of a joint trellis for DBPSK.Starting from the GMSK example, generalisation to other forms of CPM isstraightforward. Similarly, it is straightforward to apply the samemethod to operate on the expanded trellis resulting from differentialmodulation (eg DBPSK, DQPSK or π/4-DQPSK) and the accumulator. Such anexpanded trellis structure can also be applied when additionalnon-memoryless line codes such as non-return-to-zero (NRZ) ornon-return-to-zero-inverted (NRZI) are used. The soft bits {tilde over(d)}_(k) ^((l)) 23 are passed to the LDGM soft decoder (discussed below)which then updates its soft bit estimates {tilde over (d)}_(k) ^((l+1))24. This iterative process continues until a stopping criteria isreached. At this point the decoder can output final estimates of theoriginal plurality of information bits, û_(i) 26.

In a further embodiment, it was further realised that a recursivemodulator 21 can fulfil the requirement for an inner recursive code, andthus the accumulator of the IRA encoder 5 can be replaced with anyrecursive modulator. That is the transmitter can be further simplifiedby eliminating the accumulator altogether and replacing it with arecursive modulator 21. A functional block diagram of a transmitter andreceiver for encoding and decoding a Low Density Generator Matrix (LDGM)code with recursive modulation according to this embodiment isillustrated in FIG. 4. As illustrated in FIG. 4, the transmittercomprises a serial concatenation of a LDGM encoder 4 directly with arecursive modulator 21 (eliminating the standard accumulator 5,interleaver 11, and modulator 12 with a recursive modulator of prior artsystems). This yields a convenient, low complexity encoder/modulatorstructure that can be iteratively demodulated/decoded by a receiver.Similarly the receiver is simplified as only a soft demodulator 30 isrequired prior to the LDGM soft decoder 25 (compare FIG. 4 with FIGS. 2and 3).

Iterative decoding proceeds as described above, replacing the softdecoder for the joint recursive modulator/accumulator 22 with a softdecoder for the recursive modulator 30 alone. An example for use with aGMSK (ie CPM recursive modulation) scheme is described below. Startingfrom the GMSK example, generalisation to other forms of CPM isstraightforward. The soft decoder for differential modulation (eg DBPSK,DQPSK or π/4-DQPSK) works in a similar fashion, operating on the trellisdefined by the differential modulation scheme in use (see below forexamples of these trellises).

In some cases, the decoder/demodulator is itself a component of a largersystem. For example it may be used in a multiple access systemconsisting of many users, each independently transmitting encoded andmodulated data. In such scenarios, it is also desirable to obtain, inaddition to the decoded bits, a soft estimate of the transmitted signal31 from the soft demodulator as is illustrated in FIG. 5. This can thenbe used to (iteratively) cancel interference to improve the performanceof other users. Another application is turbo equalisation. A signalreconstruction could be achieved by taking a hard decision on the sourcebits and remodulating the result. However, there are several problemswith this approach. Firstly, if the decoded bit error rate is poor, weend up with a very bad estimate of the transmitted signal, which canincrease, rather than decrease the amount of interference in aniterative interference cancellation application. Secondly, withrecursive modulation such as differential modulation or CPM, a singlebit error or erasure can lead to a completely incorrect signal estimatefollowing that bit. Again, this catastrophically increases interferencein a cancellation receiver. The methods and system described herein canalso be further adapted to address these problems by reconstructing asoft signal estimate using all available information from the improveddecoder described herein.

FIG. 5 is a functional block diagram of a soft demodulator 30 forgeneration of soft estimate of a transmitted signal 31 according to anembodiment. This soft demodulator 30 may be used to replace the softdemodulators shown in either FIG. 3 or 4 (ie to additionally produce asoft estimate of the transmitted signal 31). The soft estimate isobtained by using all of the available information, namely the currentvalues of the soft bits {circumflex over (d)}_(k) ^((l)) 24 receivedfrom the LDGM decoder as well as the received signal y(t) 19. Again, anexample for use with a GMSK (ie CPM recursive modulation) scheme isdiscussed below, and extension to other types of CPM is straightforward.The soft remodulator for differential modulation is described below.

FIG. 6 is a functional block diagram of a transmitter 33 and iterativereceiver 34 for encoding and decoding a Low Density Generator Matrix(LDGM) code with recursive modulation according to one embodiment. FIG.6 thus generalises the embodiments illustrated in FIGS. 3, 4 and 5. Thetransmitter consists of a serial concatenation of a low-densitygenerator matrix encoder 4 and either (a) an accumulator 5 followed by arecursive modulator 21 (as illustrated in FIG. 3) or (b) just arecursive modulator 21 (as illustrated in FIG. 4). The soft demodulator30 is also configured to additionally produce a soft estimate of thetransmitted signal 31.

A detailed embodiment in which the recursive modulator 21 is a Gaussianminimum shift keying (GMSK) modulator (ie an example of a CPM modulationscheme) will now be described. We first describe the LDGMencoder/decoder 4/25, followed by the details of the GMSK modulator anddemodulator. The iterative decoder exchanges soft information 23, 24,between component decoders for the combined accumulator/GMSK (or GMSKalone) and for the LDGM code. Given the examples below, implementationsfor other systems where the GMSK components are replaced with any otherform of recursive modulation may be readily derived by those skilled inthe art. Further embodiments in which the GMSK modulator (andcorresponding demodulator) in FIG. 6 is replaced by a differentialmodulator for implementing differential binary phase shift keying(DBPSK), differential quaternary phase shift keying (DQPSK), or someother variant such as π/4 offset differential quaternary phase shiftkeying (π/4-DQPSK) will then be discussed.

We firstly consider the LDGM Encoder 4. It is noted that omitting theaccumulator stage 5 from a non-systematic irregular repeat accumulatecode results in a non-systematic irregular repeat/irregular parity checkcode. This is completely equivalent to a linear code with a k×n lowdensity generator matrix G (ie a LDGM code). Let u=(u₁, . . . , u_(k))be the k information bits arranged in a length k row vector. Then thecorresponding code-word of n bits, arranged as a length n row vectorv=(v₁, . . . , v_(n)) is obtained via the matrix-vector multiplicationv=uG. The binary matrix G is sparse, having a relatively low density ofones compared to zeros. The Hamming weight l_(i) of row i=1, 3, . . . ,k corresponds to the number of times bit u_(i) is repeated. Thepositions of the ones in row i determine the interleaving applied to biti. Similarly, the Hamming weight r_(j) of column j=1, 2, . . . , n isthe number of bits participating in parity check j. Since G is sparse,the vector-matrix multiplication can be implemented with low complexityby a person skilled in the art.

Each code bit v_(i)ε{0,1} is mapped to an antipodal signald_(i)ε{−1,+1}, eg via 0

+1, 1

−1. Alternatively, groups of bits may be mapped onto higher ordersymbols (eg QPSK or 8PSK). Different choices exist for these mapping,including Gray mapping, natural mapping, and the mappings describedbelow in relation to differential modulation. Whilst any mapping may beused, the performance of the system will vary based upon the choice ofmapping.

The LDGM code is decoded on the factor graph of the code defined by thegenerator matrix. G, consisting of variable nodes for the informationbits, variable nodes for the code bits, and check nodes. With referenceto FIG. 6, the inputs to the decoder are a-priori log likelihood values(L-values) of the code bits, {tilde over (d)}_(k), k=1, 2, . . . , n,and the outputs are the extrinsic L-values of the code bits, {circumflexover (d)}_(k), k=1, 2, . . . , n. After a certain number of iterationsor after a stopping criterion is fulfilled, the a-posteriori L-values ofthe information bits are computed, and based on that, the estimates ofthe source bits, û_(i), i=1, 3, . . . , k, are determined. Suitablestopping criterion may be based on a percentage change in one or moreparameters (eg estimates of the source bits or L-values) indicating astable estimate has been reached, or a threshold level being reached.For example if one, a percentage (eg 50%, 75%, 90%) or all of theposteriori L-values meeting/passing a threshold value. Any messagepassing algorithm may be applied to perform the computation. Suitableexamples include the sum-product algorithm, min-sum algorithm, or evenbinary-message passing algorithms as would be known to the personskilled in the art. They may also be based upon a hard decisionestimate.

In this embodiment, the modulator is a GMSK modulator. The GMSKmodulator maps the sequence of binary symbols d_(n)ε{−1,+1}, n=0, 1, . .. , N−1, to the continuous phase signal x(t). The modulation parametersare: symbol duration T, 3 dB bandwidth B (or equivalently the productBT), pulse length L symbols (ie, pulse duration LT), and transmit energyE_(s) per symbol.

The GMSK signal can be derived as follows. For tε[0,(N+L)T), the complexbaseband signal is

$\begin{matrix}{{x(t)} = {\sqrt{\frac{E_{s}}{T}}^{j\; {\phi {(t)}}}}} & (1)\end{matrix}$

with the information-bearing phase

$\begin{matrix}{{{\phi (t)} = {\phi_{0} + {\pi {\sum\limits_{n = 0}^{N - 1}{d_{n} \cdot {q\left( {t - {nT}} \right)}}}}}},} & (2)\end{matrix}$

where φ₀ is a constant phase offset and q(t) is the causal GMSK phasepulse (see below). Notice that ∫₀ ^(T)∥e^(jφ(t))∥²dt=T. The GMSK phasepulse has the properties

$\begin{matrix}{{q(t)} = \left\{ \begin{matrix}0 & {for} & {t \leq 0} \\{{monotonically}\mspace{14mu} {increasing}} & {for} & {0 \leq t \leq {LT}} \\{1/2} & {for} & {{LT} < t}\end{matrix} \right.} & (3)\end{matrix}$

To define the GMSK, consider first the time-symmetric non-truncatedphase pulse q₀(t) and the corresponding frequency pulse q′₀(t). Thephase pulse q₀(t) is defined as the integral of the frequency pulseq′₀(t),

q ₀(t)=∫_(−∞) ^(t) q′ ₀(τ)dτ;  (4)

the frequency pulse is defined as the convolution of the rectangular MSKpulse r(t) with a Gauss pulse g(t),

q′ ₀(t)=r(t)*g(t),  (5)

where

$\begin{matrix}{{r(t)} = \left\{ \begin{matrix}\frac{1}{2T} & {{{for}\mspace{14mu} - \frac{T}{2}} \leq t \leq \frac{T}{2}} \\0 & {otherwise}\end{matrix} \right.} & (6) \\{{g^{(i)} = {\frac{1}{\sqrt{2\pi}\sigma_{g}}^{- \frac{t^{2}}{2\sigma_{g}^{2}}}}},{\sigma_{g} = {\frac{T\sqrt{\ln \; 2}}{2\pi \; {BT}}.}}} & (7)\end{matrix}$

The pulses can be expressed analytically using g(t) and Q(x):

$\begin{matrix}{\mspace{20mu} {{q_{0}^{\prime}(t)} = {\frac{1}{2T}\left\lbrack {{Q\left( \frac{t - {T/2}}{\sigma_{g}} \right)} - {Q\left( \frac{t + {T/2}}{\sigma_{g}} \right)}} \right\rbrack}}} & (8) \\{{q_{0}(t)} = {\frac{1}{2} + {{{\frac{\sigma_{g}^{2}}{2T}\left\lbrack {{g\left( {t + {T/2}} \right)} - {g\left( {t - {T/2}} \right)}} \right\rbrack}++}{\frac{t - {T/2}}{2T} \cdot {Q\left( \frac{t - {T/2}}{\sigma_{g}} \right)}}} - {\frac{t + {T/2}}{2T} \cdot {{Q\left( \frac{t + {T/2}}{\sigma_{g}} \right)}.}}}} & (9)\end{matrix}$

where the Q-function is defined as

$\begin{matrix}{{Q(t)} = {{\frac{1}{\sqrt{2\pi}}{\int_{t}^{\infty}{^{{- z^{2}}\text{/}2}{z}}}} = {\frac{1}{2}{{{erfc}\left( {t\text{/}\sqrt{2}} \right)}.}}}} & (10)\end{matrix}$

Truncating q₀(t) at −LT/2 and LT/2 and shifting it by LT/2 gives thecausal phase pulse q(t).

The GMSK modulator may be interpreted as a finite state machine. Thisdescription may be applied to implement the modulator and can be used todefine the trellis in the soft-output decoding algorithm. Consider thephase for the time interval t=kT+τ, τε[0,T), referred to as time k inthe following, and split it up as follows:

$\begin{matrix}\begin{matrix}{{\phi \left( {{kT} + \tau} \right)} = {\phi_{0} + {\pi {\sum\limits_{n = 0}^{k}{d_{n} \cdot {q\left( {\tau + {\left( {k - n} \right)T}} \right)}}}}}} \\{= {\phi_{0} + {\frac{\pi}{2}{\sum\limits_{n = 0}^{k - L}d_{n}}} + {\pi {\sum\limits_{n = {k - L + 1}}^{k}{d_{n} \cdot {q\left( {\tau + {\left( {k - n} \right)T}} \right)}}}}}} \\{{= {\phi_{0} + \theta_{k} + {\pi {\sum\limits_{i = 0}^{L - 1}{d_{k - i} \cdot {q\left( {\tau + {i\; T}} \right)}}}}}},}\end{matrix} & (11) \\{with} & \; \\{\theta_{k} = {{\frac{\pi}{2}{\sum\limits_{n = 0}^{k - L}d_{n}}} = {\theta_{k - 1} + {\frac{\pi}{2}{d_{k - L}.}}}}} & (12)\end{matrix}$

The pulses associated to d_(n), n≦k−L, have reached their maximum value½, and their contributions to the phase are gathered in θ_(k). Noticethat

$\theta_{k} \in \left\{ {0,\frac{\pi}{2},\pi,\frac{3\pi}{2}} \right\}$

and in particular

$\begin{matrix}{\theta_{k} \in \left\{ \begin{matrix}\left\{ {0,\pi} \right\} & {{if}\mspace{14mu} k\mspace{14mu} {is}\mspace{14mu} {even}} \\\left\{ {\frac{\pi}{2},\frac{3\pi}{2}} \right\} & {{if}\mspace{14mu} k\mspace{14mu} {is}\mspace{14mu} {odd}}\end{matrix} \right.} & (13)\end{matrix}$

As can be seen from (11), the phase in this time interval depends onlyon the state

$\begin{matrix}{{s_{k} = {\text{(}\theta_{k}}},\underset{\underset{L - {1\mspace{11mu} {symbols}}}{}}{\left. {d_{k - L + 1},d_{k - L + 2},\ldots \mspace{11mu},d_{k - 2},d_{k - 1}} \right)}} & (14)\end{matrix}$

and the current symbol d_(k). The state space is given by

$\begin{matrix}{{S = {\left\{ {0,\frac{\pi}{2},\pi,\frac{3\pi}{2}} \right\} \times \left\{ {{- 1},{+ 1}} \right\}^{L - 1}}},} & (15)\end{matrix}$

and has size |S|=4·2^(L−1)=2^(L+1). Note that the size of the statespace can be reduced by a factor of 2 by considering (13).

Thus we have a finite state machine representation 70 of the GMSKmodulator, which is illustrated in FIG. 7. Define the initial state ass₀=(θ₀,d_(−L+1), . . . , d⁻¹) with (the arbitrary choice of)d_(−L+1)=d_(−L+2)=d⁻¹=−1 and θ₀=0. Also arbitrarily, we may choose theinitial phase as φ₀=0. We define the phase function for one symbolduration T as

$\begin{matrix}{{{\phi_{T}\left( {{\tau;s},d} \right)} = {\phi_{0} + \theta + {\pi {\sum\limits_{i = 0}^{L - 1}{d_{- i}^{\prime} \cdot {q\left( {\tau + {i\; T}} \right)}}}}}},{\tau \in \left\lbrack {0,T} \right)}} & (16)\end{matrix}$

for state s=(θ,d′_(−L+1), . . . , d′⁻¹) and d′₀=d₀. Then for t=kT+τ,τε[0,T),

φ(kT+τ)=φ_(T)(τ;s _(k) ,d _(k)).  (17)

Similarly, we define the normalised signal function (with unit energy)for one symbol duration T as

$\begin{matrix}{{x_{T}\left( {{\tau;s},d} \right)} = {\frac{1}{\sqrt{T}}{^{{j\phi}{({{\tau;s},d})}}.}}} & (18)\end{matrix}$

The set of all such signal functions is denoted by

={x _(T)(τ;s,d):(s,d)ε≡×{−1,+1}}.  (19)

Then for t=kT+τ, τε[0,T),

x(kT+τ)=√{square root over (E _(s))}·x _(T)(τ;s _(k) ,d _(k)).  (20)

These relations are also illustrated in FIG. 7. The states 71 defined in(14) are shown in the top of the diagram and each state is provided tothe waveform lookup table 72. The recursive nature of the GMSK modulatoris apparent from the generation of states 71 in the top of FIG. 7, inwhich element D indicates a delay element (ie for one symbol duration)that is equivalently a memory that stores the previous state. Thewaveform lookup table stores the waveforms associated with each of thestates given by (15) to obtain the output phase given by (17) from whichthe normalised signal function 73 can be obtained by (18). Thenormalised signal 73 is then multiplied by √{square root over (E_(s))}to obtain the output signal 74 given by (20).

A GMSK Soft Demodulator can also be developed. The GMSK signal may berepresented in a trellis with states, as defined above. The transitionfrom current state s_(k) with current input symbol d_(k) to next states_(k+1) is given by

$\begin{matrix}{s_{k} = {{\left( {\theta_{k},d_{k - L + 1},\ldots \mspace{11mu},d_{k - 1}} \right)\overset{d_{k}}{\rightarrow}s_{k + 1}} = \left( {\theta_{k + 1},d_{k - L + 2},\ldots \mspace{11mu},d_{k}} \right)}} & (21)\end{matrix}$

with θ_(k+1)=θ_(k)+(π/2)d_(k−L+1). Thus, GMSK can be decoded on thistrellis with the Viterbi algorithm, the BCJR algorithm, or anyvariations thereof in the standard way, where also a-priorisoft-information (probabilities) of the input symbols may be used.

Generalisation to other memoryless channels is straightforward, and tochannels with memory is possible. Assume transmission of the signal x(t)over an Additive White Gaussian Noise (AWGN) channel. The channel outputis

y(t)=x(t)+w(t),  (22)

where w(t) is a complex white Gaussian noise process with spectral powerdensity N₀ (N₀/2 per dimension). The GMSK demodulator obtains thechannel output y(t) and a-priori information {circumflex over (d)}_(k)for each transmitted symbol d_(k), k=0, 1, . . . , N+L. Without loss ofgenerality, we assume that this information is provided in terms oflog-likelihood ratios (L-values), ie,

$\begin{matrix}{{\hat{d}}_{k}:={{L\left( d_{k} \right)} = {\log {\frac{P\left( {d_{k} = {+ 1}} \right)}{P\left( {d_{k} = {- 1}} \right)}.}}}} & (23)\end{matrix}$

Using these input values, the GMSK demodulator computes the extrinsicL-values of the symbols d_(k),

$\begin{matrix}{{{\overset{\sim}{d}}_{k}:={{L_{c}\left( d_{k} \right)} = {\log \frac{P\left( {{d_{k} = {{+ 1}{y(t)}}},{\hat{d}}_{0},\ldots \mspace{11mu},{\hat{d}}_{k - 1},{\hat{d}}_{k + 1},\ldots \mspace{11mu},{\hat{d}}_{N + L}} \right)}{P\left( {{d_{k} = {{- 1}{y(t)}}},{\hat{d}}_{0},\ldots \mspace{11mu},{\hat{d}}_{k - 1},{\hat{d}}_{k + 1},\ldots \mspace{11mu},{\hat{d}}_{N + L}} \right)}}}},} & (24)\end{matrix}$

k=0, 1, . . . , N+L, using standard trellis-based algorithms, like BCJR,LogAPP, SOVA, etc. The branch metric to be used in these algorithmscomprises the a-priori L-values and the log-likelihoods of thewaveforms. The latter is considered in the following.

An optimal method of demodulation is to correlate the received signalwith all possible transmitted waveforms for time intervals [kT,(k+1)T).This may be written using the real part of the inner product,

b _(k)({tilde over (s)},{tilde over (d)})=

{

y(kT+τ),x _(T)(τ;{tilde over (s)}{tilde over (d)})

},  (25)

where ({tilde over (s)},{tilde over (d)})ε≡×{−1,+1} denotes thehypothesis and the inner product is defined as

x(t),y(t)

=∫₀ ^(T) x(t)y*(t)dt  (26)

for signals confined to the interval tε[0,T) and (.)* denotes complexconjugation. We have for the signal and the noise part

x(t;{tilde over (s)},{tilde over (d)}),x _(T)(τ;{tilde over (s)},{tildeover (d)})

=√{square root over (E _(s))}

w(t),x _(T)(τ;{tilde over (s)},{tilde over (d)})

=0

|

w(τ),x _(T)(τ;{tilde over (s)},{tilde over (d)})

|² =N ₀,  (27)

and thus the log-likelihood is given by

$\begin{matrix}{{{{logp}\left( {{y\left( {{kT} + \tau} \right)}{x_{T}\left( {{\tau;\overset{\sim}{s}},\overset{\sim}{d}} \right)}} \right)} = {{2{\frac{E_{s}}{N_{0}} \cdot {b_{k}\left( {\overset{\sim}{s},\overset{\sim}{d}} \right)}}} + c_{0}}},} & (28)\end{matrix}$

where c₀ is a constant independent of the transmitted waveform, and canthus be ignored.

Combining this log-likelihood with the a-priori L-values for the symbolsgives the desired additive branch metric

$\begin{matrix}{{\mu_{k}\left( {\overset{\sim}{s},\overset{\sim}{d}} \right)}:={{L\left( d_{k} \right)} + {2{\frac{E_{s}}{N_{0}} \cdot {{b_{k}\left( {\overset{\sim}{s},\overset{\sim}{d}} \right)}.}}}}} & (29)\end{matrix}$

Alternatively, the branch metrics may be computed using the amplitudemodulation (AM) representation of CPM signals. In another embodiment,demodulation is performed by correlating the received signal with a setof approximations of the transmitted waveforms. In one embodiment, thisset of exact or approximate waveforms are linear combinations of pulseamplitude modulated (PAM) waveforms obtained from a Laurentdecomposition of the CPM signal.

GMSK Soft Signal. Reconstruction can also be performed. The GMSKdemodulator can also be used to reconstruct a soft signal estimate basedon the noisy observation y(t) of the signal x(t) and the a-prioriinformation {circumflex over (d)}_(k) of the symbols d_(k). Inparticular, we aim to compute the expectation (

[.])

{circumflex over (x)}(t)=

[x(t)|y(t),{circumflex over (d)} ₀ ,{circumflex over (d)} ₁ , . . .,{circumflex over (d)} _(N+L)].  (30)

This type of signal estimate is typically used in multi-user receiversfor soft interference cancellation. Consider the estimate {circumflexover (x)}(kT+τ), τε[0,T), for one symbol period. We can compute theprobability

q _(k)(x _(T)(τ))=P({circumflex over (x)}(kT+τ)=x_(T)(τ)|y(t),{circumflex over (d)} ₀ ,{circumflex over (d)} ₁ , . . .,{circumflex over (d)} _(N+L))  (31)

for each waveform x_(T)(τ)εχ, cf. (19), and all k=0, 1, . . . , N+L, onthe GMSK trellis with a trellis-based algorithm, eg the BCJR LogAPPalgorithm, SOVA, etc. The desired expectation is then given by theaverage:

$\begin{matrix}{{{\hat{x}\left( {{kT} + \tau} \right)} = {\sum\limits_{{x_{T}{(\tau)}} \in X}{{q_{k}\left( {x_{T}(\tau)} \right)} \cdot {x_{T}(\tau)}}}},} & (32)\end{matrix}$

where k=0, 1, . . . , N+L.

As was described above, and as was illustrated in FIG. 3, in oneembodiment, a combined accumulator and recursive modulator may also beused to replace the accumulator/interleaves/modulator architecture ofthe prior art. As will be shown using a joint accumulator/GMSK modulatorinstead of the plain GMSK modulator described above implies only minorchanges in the modulator and the demodulator.

The joint modulator is obtained by precoding the GMSK modulator with anaccumulator, or equivalently by a differential precoder, as illustratedin FIG. 8 which is a finite state machine representation 80 of a jointaccumulator and GMSK modulator according to an embodiment.

For convenience, we keep the notation for the GMSK modulator, and denotethe input to the accumulator by d′_(k) 81. Though the accumulator seemsto introduce a new delay element 82 and thus additional memory, itactually does not. The symbol d_(k−1) 83 required for the accumulatoroperation

d _(k) =d′ _(k) ·d _(k−1)  (33)

is already available in the memory of the GMSK modulator (note that inchannel coding literature, the accumulator is usually described withsymbols from {0,1} and binary addition, which is equivalent to ourdescription in which 0

+1, and 1

−1). Therefore, the trellis complexity of the combinedaccumulator/GMSK-modulator is actually exactly the same as thecomplexity of the GMSK modulator described above. Even more, the trellisof the combined accumulator/GMSK-modulator may easily be obtained byrelabelling the GMSK-modulator by a person skilled in the art. Thealgorithms for demodulation and soft signal reconstruction are identicalto those for the (plain) GMSK modulator, only the labels of the trellistransitions are different.

The above description provides an example of how to implement an LDGMcode with reduced implementation complexity for use with continuousphase modulation schemes such as GMSK. This work will now be extended toconsider differential modulation schemes such as differential binaryphase shift keying (DBPSK). As was the case for CPM schemes, we willfirst consider the case in which the accumulator, interleaver andmodulator are replaced directly with a recursive modulator 21.

FIG. 9 is a finite state machine representation 90 of a Differential.Binary Phase Shift Keying (DBPSK) modulator according to an embodiment.At the input, binary symbols d_(k)ε{−1,1} 91 are modulated 92 ontobinary phases ψ_(B)(d_(k))ε{0,π} 93 (this is a BPSK modulator). Theoutput signal θ_(k) 95, which is the phase to be transmitted at time k,is formed as

θ_(k)=θ_(k−1)⊕ψ(d _(k)),  (34)

where the addition 94 of the previous phase 97 (obtained via delayelement D or memory) is modulo 2π (ie wraps around at 2π, so thatm2π+θ≡θ for integer m). Note that the structure of the DBPSK modulatoris similar to that of the accumulator.

It is common practice to represent a finite state machine using atrellis diagram. FIG. 10 shows the finite state machine 90 of the DBPSKmodulator of FIG. 9 represented as a trellis 100. The states arelabelled with the corresponding values of the previous phase θ_(k−1) 101and current phase θ_(k) 102. The branch labels 103 are the values of theinput d_(k) which cause the corresponding state transitions.

Assuming rectangular pulse shaping (other pulse shapes can be easilyincorporated), the transmitted signal for the interval of duration Tcorresponding to state s=θ and symbol d is

$\begin{matrix}{{x_{T}\left( {{\tau;s},d} \right)} = {\frac{1}{\sqrt{T}}{^{{j\phi}{({\tau,s,d})}}.}}} & (35)\end{matrix}$

where

φ(τ;θ,d)=θ⊕ψ_(B)(d),τε[0,T).  (36)

Other common types of differential modulation are differentialquaternary phase shift keying (DQPSK), a finite state machinerepresentation 110 of which is shown in FIG. 11, and π/4 offset DQPSK, afinite state machine representation 130 of which is shown in FIG. 13.These differ in the set of phase shifts that are used to encode the twobits of data.

For DQPSK (FIG. 11), each pair of bits 111 is encoded 112 into a phaseshift 113 from 0,π/2,π,3π/2 with respect to the previous symbol. Onepossible mapping is shown in FIG. 11, other mappings are also possible.The input binary symbols are selected fromd_(k)ε{00,01,11,10}≡{+1+1,+1−1,−1−1,−1+1} (based upon 0

+1, and 1

−1). The possible output phases 115 are from the set 0,π/2,π,3π/2. Againthe output phase is obtained from the addition of the previous phase 117with phase shift 113.

The trellis representation 120 of DQPSK is shown in FIG. 12. The trellisshows the previous phase θ_(k−1) 121 and current phase θ_(k) 122, andnow has four states and is fully connected. The branches are labelled123 according to the value of the input d_(k) causing that transition(with −1 abbreviated by − and +1 abbreviated by +). The output signal isobtained in the same way as (35), (36) with ψ_(Q)(d) replacing the BPSKmodulator (according to FIG. 11).

For π/4 offset DQPSK (FIG. 13), the allowable phase shifts 133 are±π/4,±3π/4 (again, one example of mapping 132 is given in the figure,but others are possible). This is to avoid phase changes of π (ie signchanges), which can be undesirable. There are now eight possible outputphases, 0,±π/4,π,±π/2,±3π/4 135, however, as shown in FIG. 14 which isthe trellis diagram 140 of π/4 offset DBPSK, given any phase θ_(k−1)141, the phase θ_(k) 142 is one of only four possibilities. Ifθ_(k−1)ε{0,π/2,π,3π/2}, then θ_(k)ε{±π/4,±3π/4}. If θ_(k−1)ε{±π/4,±3π/4}then θ_(k)ε{0,π/2,π,3π/2}. Again, the branch labels 143 are the valuesof the input d_(k) which cause the corresponding state transitions. Theoutput signal is obtained in the same way as (35), (36) with ψ_(Q′)(d)replacing the BPSK modulator (according to FIG. 13).

As was the case for CPM schemes, the above theory may be applied to analternative embodiment in which the accumulator, interleaver andmodulator are replaced with a joint accumulator and differentialmodulator. It is possible to make a joint trellis for the combination ofthe accumulator followed by differential modulation. FIG. 15 is atrellis representation 150 of a joint accumulator/DBPSK. The states arelabelled with the pair d_(k−1), θ_(k−1) (previous state) 151 and d_(k),θ_(k) (current state) 152. The transitions are labelled 153 with thevalue of d_(k) that causes the transition. From the diagram, we see thatthere are now four states (storing not only the previous data bit, butalso the previous phase). However, there are only two possibletransitions from each state (one for each value of d_(k)). Thus theoverall complexity of implementing a soft decoder (eg BCJR and itsvariants) is the same as that for DBPSK alone. It is straightforward forsomeone skilled in the art to derive the corresponding joint trellisesfor other forms of differential modulation, such as DQPSK and π/4-DQPSK.

As was already described above in relation to CPM and GMSK, a softdemodulator for differential modulation can also be implemented usingthe well-known BCJR algorithm or any of its variants, such as LogAPP,SOVA etc. The computation of the branch metrics in this case follows thesame principle as described in relation to the GMSK soft demodulator,where we use the appropriate definition of x_(T)(τ;s,d) for recursivemodulation (ie equation (35) rather than equation (18)).

Similarly, soft signal reconstruction for differential modulation can beachieved using the method described in relation to GMSK soft signalreconstruction. We again replace the waveforms x_(T)(τ) with theappropriate definition in relation to recursive modulation.

The performance of an embodiment of the LDGM-coded GMSK modulation willnow be described. FIG. 16 is a plot 160 of the bit error rate (BER) forthe transmitted data bits as a function of the signal-to-noise ratio,and FIG. 17 is a plot 170 the mean square error (MSE) between thetransmitted signal and the remodulated signal as a function of thesignal-to-noise ratio.

Consider GMSK with bandwidth-time product BT=0.4, and pulse length L=5symbols and transmission over an AWGN channel with signal-to-noise ratioE_(s)/N₀. The LDGM code has rate ½ and its generator matrix is randomlyconstructed from an ensemble with degree polynomialsλ(z)=0.1z¹+0.1z²+0.8z⁷ and ρ(z)=0.1+0.2z+0.7z². The LDGM code is decodedon the factor graph with 10 iterations per decoder activation. For thereceiver with iterations between the GMSK demodulator and the LDGMdecoder, we assume 10 iterations as well. Decoding is terminated if theestimated data bits at the decoder output fulfil a CRC. However, it isto be understood that this may be replaced by other terminationcriteria, for example based upon a maximum number of iterations, alikelihood value or change (or lack of) in a likelihood value or otherparameter.

FIG. 16 compares the BER of the three systems. The first system isuncoded GMSK (ie no LDGM) which provides a reference level shown as theline with circles 161. The second system is LDGM-coded. GMSK withoutiterations between the demodulator and the decoder (referred to as“non-iterative” in the figures), shown as the line with triangles 162.That is, a hard estimate of the bits are provided to the decoder fromthe soft demodulator. This second system is a modification of the systemshown in FIG. 4 (ie without iteration between LDGM Sof Decoder 25 andthe Soft Demodulator 31). The third system is LDGM-coded GMSK withiterations between the demodulator and the decoder, shown as the linewith squares 163. This last system illustrated the performance of thesystem shown in FIG. 4.

At a BER of 10⁻⁵, the coded system with iterations outperforms theuncoded system by over 8 dB. Further the non-iterative coded systemstill has a gain of about 3.5 dB over the uncoded system.

The benefits of soft remodulation versus hard remodulation aredemonstrated in FIG. 17 and discussed below. We apply the MSE betweenthe transmitted signal and the remodulated signal to measure the qualityof the signal reconstruction. The following four remodulation schemesare considered.

In the first scheme, the receiver operates non-iteratively (ie withoutiterations between the demodulator and decoder) and makes hard decisionson the transmitted data bits. If the CRC is fulfilled, these harddecisions are encoded by the channel code and the code-word is fed tothe GMSK modulator to reconstruct the transmitted signal. If the CRCfails, then typically a signal reconstructed based on these erroneousdata bits does not have much in common with the original signal,therefore in this case, the reconstructed signal is simply set to be theall-zero signal, corresponding to no reconstruction, shown as line 171.

In the second scheme, the receiver operates non-iteratively, however,the soft remodulation method (including the GMSK soft signalreconstruction) is applied to reconstruct the transmitted signal, shownas line 172. That is the soft demodulator shown in FIGS. 5 and 6 thatgenerates the soft estimate of the transmitted signal 31 is used.

In the third scheme the receiver operates iteratively (ie withiterations between the demodulator and the decoder, as is shown in FIG.4). Otherwise the operation is the same as the first scheme, shown asline 173.

In the fourth scheme, the receiver operates iteratively and the softremodulation method (including the GMSK soft signal reconstruction) isapplied to reconstruct the transmitted signal, shown as line 174. Thatis the soft demodulator shown in FIGS. 5 and 6 that generates the softestimate of the transmitted signal 31 is used.

For all four schemes, we reconstructed the signal based on harddecisions if the channel decoder indicated error free decoding (based onthe CRC check), in which case the MSE is zero. The simulation resultsare shown in FIG. 17. As expected the systems perform in the order aslisted above (171, 172, 173, 174), with the last one giving the bestperformance. In can also be seen that the soft signal reconstructionalways outperforms the hard signal reconstruction.

For the iterative receivers, the soft remodulation has an MSE of about ½at low SNR, while the hard remodulation has an MSE of about 1. This hasa big effect in iterative multi-user decoding. While the softremodulator decreases the interference by about 3 dB and thus starts theiterative process, the hard demodulator has no effect at all.

The performance of the overall system may be improved by optimising thedegree polynomials of the LDGM code, the number of decoder iterations,and the number of iterations between the demodulator and the decoder forany given GMSK modulation.

FIG. 18 is a flowchart of a method 180 for generating a signal fortransmission in a communication system using a recursive modulationscheme and a Low Density Generator Matrix (LDGM) code. The methodcomprises:

encoding a plurality of received source bits and generating a pluralityof coded symbols, wherein the encoder includes Low Density GeneratorMatrix (LDGM) encoder to generate a plurality of coded symbols accordingto a LDGM code (step 182); and

modulating the plurality of coded symbols using a recursive modulationscheme to generate a signal for transmission (step 184).

FIG. 19 is a flowchart of a method 190 for receiving a transmission in acommunication system using a recursive modulation scheme and a LowDensity Generator Matrix (LDGM) code. The method comprises:

soft demodulating a received transmit signal according to a recursivemodulation scheme to generate a plurality of coded symbol estimates(192); and

decoding the plurality of coded symbol estimates using a Low DensityGenerator Matrix (LDGM) decoding scheme and generating a plurality ofsource bit estimates (194).

The coded symbol estimates and source bit estimates may be hard or softestimates. The soft demodulating and soft decoding may be performediteratively until a stopping criterion is reached, wherein during eachiteration, the soft demodulation is performed based upon a plurality ofsoft a-priori estimates of the coded symbol estimates received from theprevious decoding step and are used to generate updated coded symbolestimates for use as soft a-priori estimates in the next iteration; anddecoding the plurality of coded symbol estimates is performed based upona plurality of soft a-priori coded symbol estimates received from theprevious soft demodulation step and are used to generate updated codesymbol estimates for transmission to the soft demodulator for use asa-priori coded symbol estimates in the next iteration.

These methods may be performed by relevant functional blocks withintransmitter and receiver systems. For increased clarity and ease ofdiscussion other common functional blocks have been omitted such as datasources, framing units, controllers/processors, memory units, CRCcheckers, transmitter and receiver units which generate/receive theactual signal. For example a transmitter unit may process the modulationsymbols in accordance with the design of the system and generates datasamples and further conditions (eg, converts to analog, filters,amplifies, and frequency up-converts) the data samples to generate amodulated signal, which is transmitted via an antenna. Equivalentfunctions may be performed by the receiver. A controller or processormay be used to control the various functional blocks. It will also beunderstood that the methods described herein may be used for any digitalcommunication system, whether wired or wireless, which is designed touse a Low Density Generator Matrix code with a recursive modulationscheme. Wireless systems may be RF based systems such as those operatingin the UHF and VHF ranges. Wired systems may be cable or optical fibresystems. The various coding and modulation schemes may be implemented inhardware or circuits using one or more Field Programmable Gate Arrays(FPGA), Digital Signal Processors (DSP), or other appropriate hardware.The transmitters and receivers may further comprise other circuits ormodules such as filters, equalisers, mixers, up-converters,down-converters, etc, for generating or processing signals including thetransmit and receive signal.

Embodiments of a method and system for implementing a Low DensityGenerator Matrix code including an irregular repeat accumulate (IRA)code for use with recursive modulation schemes which has reducedcomplexity and thus reduced cost have been described herein. In oneembodiment illustrated in FIG. 3, a combined accumulator and recursivemodulator is used. In this embodiment, the transmitter comprisesconcatenation of a low density generator matrix code, followed by anaccumulator and recursive modulation, and a corresponding receiver. Thisaspect eliminates the additional interleaver and combines the decoderfor the accumulator and the soft demodulator in the receiver into asingle joint decoder thereby reducing the number of components andcomplexity. In another embodiment, illustrated in FIG. 4, thetransmitter is further simplified by eliminated the accumulatoraltogether and replacing it with a recursive modulator. That is thetransmitter comprises serial concatenation of a low density generatormatrix code with a recursive modulator. Similarly the receiver issimplified as only a soft demodulator is required prior to the LDGM softdecoder (compare FIG. 4 with FIG. 3).

The decoder/demodulator may be a component of a larger system. Forexample the methods and apparatus may be used in a multiple accesssystem consisting of many users, each independently transmitting encodedand modulated data. In such scenarios, in addition to the decoded bits,a soft estimate of the transmitted signal can also be provided which canthen be used to (iteratively) cancel interference to improve theperformance of other users. Another application is turbo equalisation. Asignal reconstruction could be achieved by taking a hard decision on thesource bits and remodulating the result. However there are severalproblems with this approach. Firstly, if the decoded bit error rate ispoor, we end up with a very bad estimate of the transmitted signal,which can increase, rather than decrease the amount of interference inan iterative interference cancellation application. Secondly, withrecursive modulation such as differential modulation or CPM, a singlebit error or erasure can lead to a completely incorrect signal estimatefollowing that bit. Again, this catastrophically increases interferencein a cancellation receiver. The methods and system described herein canalso be further adapted to address these problems by reconstructing asoft signal estimate using all available information from the improveddecoder described herein.

The methods and receivers may be utilised in communication systems andcomponents such as those described in corresponding PCT PatentApplication (number yet to be assigned) titled “Carrier phase andamplitude estimation for phase shift keying using pilots and data” filedon 13 Dec. 2013, and the following co-pending PCT patent applications:

PCT/AU2013/000888 titled “System and Method for Analog to DigitalConversion” filed on 13 Aug. 2013;

PCT/AU2013/000895 titled “Channel Allocation in a Communication. System”filed on 14 Aug. 2013;

PCT/AU2013/001078 titled “Communication system and method” filed on 20Sep. 2013; and

PCT/AU2013/001079 titled “Multi-Access Communication System” filed on 20Sep. 2013. The content of each of these applications is herebyincorporated by reference in their entirety.

Those of skill in the art would understand that information and signalsmay be represented using any of a variety of technologies andtechniques. For example, data, instructions, commands, information,signals, bits, symbols, and chips may be referenced throughout the abovedescription may be represented by voltages, currents, electromagneticwaves, magnetic fields or particles, optical fields or particles, or anycombination thereof.

Those of skill in the art would further appreciate that the variousillustrative logical blocks, modules, circuits, and algorithm stepsdescribed in connection with the embodiments disclosed herein may beimplemented as electronic hardware, computer software, or combinationsof both. To clearly illustrate this interchangeability of hardware andsoftware, various illustrative components, blocks, modules, circuits,and steps have been described above generally in terms of theirfunctionality. Whether such functionality is implemented as hardware orsoftware depends upon the particular application and design constraintsimposed on the overall system. Skilled artisans may implement thedescribed functionality in varying ways for each particular application,but such implementation decisions should not be interpreted as causing adeparture from the scope of the present invention.

The steps of a method or algorithm described in connection with theembodiments disclosed herein may be embodied directly in hardware, in asoftware module executed by a processor, or in a combination of the two.For a hardware implementation, processing may be implemented within oneor more application specific integrated circuits (ASICs), digital signalprocessors (DSPs), digital signal processing devices (DSPDs),programmable logic devices (PLDs), field programmable gate arrays(FPGAs), processors, controllers, micro-controllers, microprocessors,other electronic units designed to perform the functions describedherein, or a combination thereof. A central processing unit (CPU) may beused, containing an Input/Output Interface, an Arithmetic and Logic Unit(ALU) and a Control Unit and Program Counter element which is incommunication with input and output devices or modules through theInput/Output Interface, and a memory. Software modules, also known ascomputer programs, computer codes, or instructions, may contain a numberof source code or object code segments or instructions, and may residein any computer readable medium such as a RAM memory, flash memory, ROMmemory, EPROM memory, registers, hard disk, a removable disk, a CD-ROM,a DVD-ROM or any other form of computer readable medium. In thealternative, the computer readable medium may be integral to theprocessor. The processor and the computer readable medium may reside inan ASIC or related device. The software codes may be stored in a memoryunit and executed by a processor. The memory unit may be implementedwithin the processor or external to the processor, in which case it canbe communicatively coupled to the processor via various means as isknown in the art.

The reference to any prior art in this specification is not, and shouldnot be taken as, an acknowledgement of any form of suggestion that suchprior art forms part of the common general knowledge, or is well-knownin the field.

Throughout the specification and the claims that follow, unless thecontext requires otherwise, the words “comprise” and “include” andvariations such as “comprising” and “including” will be understood toimply the inclusion of a stated integer or group of integers, but notthe exclusion of any other integer or group of integers.

It will be appreciated by those skilled in the art that the invention isnot restricted in its use to the particular application described.Neither is the present invention restricted in its preferred embodimentwith regard to the particular elements and/or features described ordepicted herein. It will be appreciated that the invention is notlimited to the embodiment or embodiments disclosed, but is capable ofnumerous rearrangements, modifications and substitutions withoutdeparting from the scope of the invention.

1. An apparatus for generating a signal for transmission in acommunication system implementing a recursive modulation scheme and aLow Density Generator Matrix (LDGM) code, the apparatus comprising: anencoder for receiving a plurality of source bits and generating aplurality of coded symbols, wherein the encoder includes a Low DensityGenerator Matrix (LDGM) encoder to generate a plurality of coded symbolsaccording to a LDGM code; and a recursive modulator which receives theplurality of coded symbols from the encoder and modulates the receivedplurality of coded symbols to a signal for transmission using arecursive modulation scheme.
 2. The apparatus as claimed in claim 1,wherein the encoder is an irregular repeat accumulate (IRA) code encodercomprising the LDGM code concatenated with an accumulator and theplurality of coded symbols are encoded according to an IRA code.
 3. Theapparatus as claimed in 1, wherein the recursive modulator uses adifferential encoding modulation scheme.
 4. The apparatus as claimed in1, wherein the recursive modulator uses a continuous phase modulationscheme.
 5. The apparatus as claimed in claim 1, wherein the recursivemodulator uses a finite state machine implementation.
 6. An apparatusfor receiving a transmission in a communication system implementing arecursive modulation scheme and a Low Density Generator Matrix (LDGM)code, the apparatus comprising: a soft demodulator for receiving atransmit signal and demodulating the received signal according to arecursive modulation scheme to generate a plurality of coded symbolestimates; and a Low Density Generator Matrix (LDGM) decoder forreceiving the plurality of coded symbol estimates from the softdemodulator and decoding the coded symbol estimates to generate aplurality of source bit estimates.
 7. The apparatus as claimed in claim6, wherein iterative decoding of the received signal is implementedbetween the soft demodulator and the LDGM decoder until a stoppingcriterion is reached, wherein each iteration comprises the softdemodulator receives a plurality of soft a-priori estimates of the codedsymbol estimates from the LDGM decoder, and generates updated codedsymbol estimates which are transmitted to the LDGM decoder, and the LDGMdecoder receives a plurality of soft a-priori coded symbol estimatesfrom the soft demodulator and generates updated code symbol estimateswhich are transmitted to the soft demodulator for use as soft a-prioricoded symbol estimates in the next iteration.
 8. The apparatus asclaimed in claim 7, wherein the soft demodulator further comprises anaccumulator decoder.
 9. The apparatus as claimed in claim 7, wherein thecommunication system is a multi-access system or a turbo equalizationsystem, and the soft demodulator is configured to generate a softestimate of the transmitted signal.
 10. The apparatus as claimed inclaim 6, wherein the soft demodulator implements demodulation using adifferential demodulation scheme.
 11. The apparatus as claimed in claim6, wherein the soft demodulator implements demodulation using acontinuous phase demodulation scheme.
 12. The apparatus as claimed inclaim 6, wherein the soft demodulator uses a trellis based algorithm.13. The apparatus as claimed in claim 12, wherein the soft demodulatorcorrelates the received signal with all possible transmitted waveformsover a symbol period determined using the recursive modulation scheme.14. The apparatus as claimed in claim 12, wherein the soft demodulatorcorrelates the received signal with a set of approximations of thetransmitted waveforms over a symbol period determined using therecursive modulation scheme.
 15. A communication system comprising: atransmitter further comprising: an encoder for receiving a plurality ofsource bits and generating a plurality of coded symbols, wherein theencoder includes a Low Density Generator Matrix (LDGM) encoder togenerate a plurality of coded symbols according to a LDGM code; and arecursive modulator which receives the plurality of coded symbols fromthe encoder and modulates the received plurality of coded symbols to asignal for transmission using a recursive modulation scheme; and areceiver comprising: a soft demodulator for receiving a transmit signaland demodulating the received signal according to a recursive modulationscheme to generate a plurality of coded symbol estimates; and a LowDensity Generator Matrix (LDGM) decoder for receiving the plurality ofcoded symbol estimates from the soft demodulator and decoding the codedsymbol estimates to generate a plurality of source bit estimates.
 16. Amethod comprising: generating a signal for transmission in acommunication system using a recursive modulation scheme and a LowDensity Generator Matrix (LDGM) code; encoding, using an encoder, aplurality of received source bits and generating a plurality of codedsymbols, wherein the encoder includes Low Density Generator Matrix(LDGM) encoder to generate a plurality of coded symbols according to aLDGM code; and modulating the plurality of coded symbols using arecursive modulation scheme to generate a signal for transmission. 17.The method as claimed in claim 16, wherein the encoding is performedusing an irregular repeat accumulate code comprising an LDGM codeconcatenated with an accumulator.
 18. The method as claimed in 16,wherein the encoding is performed using a differential encodingmodulation scheme.
 19. The method as claimed in 16, wherein the encodingis performed using a continuous phase modulation scheme.
 20. The methodas claimed in claim 16, wherein the modulation step uses a finite statemachine implementation.
 21. A method comprising: receiving, by areceiver, a transmission in a communication system using a recursivemodulation scheme and a Low Density Generator Matrix (LDGM) code; softdemodulating, using a soft demodulator, a received transmit signalaccording to a recursive modulation scheme to generate a plurality ofcoded symbol estimates; and decoding the plurality of coded symbolestimates using a Low Density Generator Matrix (LDGM) decoding schemeand generating a plurality of source bit estimates.
 22. The method asclaimed in claim 21, wherein the steps of soft demodulating and decodingare performed iteratively until a stopping criterion is reached, whereinduring each iteration, the soft demodulation is performed based upon aplurality of soft a-priori estimates of the coded symbol estimatesreceived from the previous decoding step and are used to generateupdated coded symbol estimates for use as soft a-priori estimates in thenext iteration; and decoding the plurality of coded symbol estimates isperformed based upon a plurality of soft a-priori coded symbol estimatesreceived from the previous soft demodulation step and are used togenerate updated code symbol estimates for transmission to the softdemodulator for use as a-priori coded symbol estimates in the nextiteration.
 23. The method as claimed in claim 22, wherein the softdemodulation step comprises a joint recursive demodulation/accumulationstep.
 24. The method as claimed in claim 21, wherein the communicationsystem is a multi-access system or turbo-equalisation system, and thesoft demodulator step further generates a soft estimate of thetransmitted signal.
 25. The method as claimed in claim 21, wherein thesoft demodulation step implements a differential demodulation scheme.26. The method as claimed in claim 21, wherein the soft demodulationstep implements a continuous phase demodulation scheme.
 27. The methodas claimed in claim 21, wherein the soft demodulation step uses atrellis based algorithm.
 28. The method as claimed in claim 27, whereinthe soft demodulation step is performed by correlating the receivedsignal with all possible transmitted waveforms over a symbol perioddetermined using the recursive modulation scheme.
 29. The method asclaimed in claim 27, wherein the soft demodulator correlates thereceived signal with a set of approximations of the transmittedwaveforms over a symbol period determined using the recursive modulationscheme.